My last post seemed to go down well so I thought I’d compile a few more images of proofs without words!

Determinant is the area of a parallelogram, by Solomon Golomb, Mathematics Magazine, March 1985.

A visual proof of Jensen’s inequality, found on Wikipedia. Jensen’s inequality states that

In the diagram, the dashed curve along the X axis is the hypothetical distribution of X, while the dashed curve along the Y axis is the corresponding distribution of Y values. Note that the convex mapping Y(X) increasingly “stretches” the distribution for increasing values of X.

I decided to start a new series called ‘Proof Without Words’: a collection of pictures which prove a mathematical fact with an image. Remember, these are in no way rigorous and are just meant to give an idea of why the fact is true!

1/2+1/4+1/8+1/16+1/32+ … = 1

Sum of the first n positive odd numbers = n^2

arctan(1) + arctan(2) + arctan(3) = π

Viviani’s Theorem

The sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle’s altitude.